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Unformatted text preview: Chapter 5 Cellular Automata A cellular automaton is a model of a world with very simple physics. “Cellular” means that the space is divided into discrete chunks, called cells. An “automaton” is a machine that performs computations—it could be a real machine, but more often the “machine” is a mathematical abstrac- tion or a computer simulation. Automata are governed by rules that determine how the system evolves in time. Usually time is divided into discrete steps, and the rules specify how to compute the state of the world during the next time step based on the current state. As a trivial example, consider a cellular automaton (CA) with a single cell. The state of the cell is an integer I will represent with the variable x i , where the subscript i indicates that x i is the state of the system during timestep i . As an initial condition, I’ll specify x = 0. Now all we need is a rule. Arbitratily, I’ll pick x i = x i- 1 + 1, which says that after each time step, the state of the CA gets incremented by 1. So far, we have a simple CA that performs a simple calculation: it counts. But this CA is a little bit nonstandard; normally the number of possible states is finite. To bring it into line, I’ll choose the smallest interesting number of states, 2, and another simple rule, x i = ( x i- 1 + 1 ) %2, where % represents the remainder (or modulus) operator. This CA performs a simple calculation: it blinks. That is, the state of the cell switches between 0 and 1 after every timestep. Most CAs are deterministic , which means that rules do not have any random elements; given the same initial state, they always produce the same result. There are also nondeterministic CAs, but I will not address them here. 5.1 Wolfram The CA in the previous was 0-dimensional and it wasn’t very interesting. 1-dimensional CAs turn out to be surprisingly interesting. 44 Chapter 5. Cellular Automata In the early 1980s Stephen Wolfram published a series of papers presenting a systematic study of 1-dimensional CAs. He identified four general categories of behavior, each more interesting than the last. To say that a CA has dimensions is to say that the cells are arranged in a contiguous space so that some of them are considered “neighbors.” In one dimension, there are three natural configurations: Finite sequence: A finite number of cells arranged in a row. All cells except the first and last have two neighbors. Ring: A finite number of cells arranged in a ring. All cells have two neighbors. Infinite sequence: An infinite number of cells arranged in a row. The rules that determine how the system evolves in time are based on the notion of a “neighborhood,” which is the set of cells that determines the next state of a given cell. Wolfram’s experiments use a 3-cell neighborhood: the cell itself and its left and right neighbors....
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- Spring '11